Creation Functions
Creation of Structures
For any nonsquare integer \(D\) congruent to \(0\) or \(1\) modulo \(4\), binary quadratic forms of discriminant \(D\) may be created. The parent object of forms of discriminant \(D\) can be created using the following commands.
Creation of Forms
Binary quadratic forms may be created by coercing a triple \([a,b,c]\) of integer coefficients into the parent structure of forms of discriminant \(D = b^2-4ac\). Other constructors are provided for constructing the group identity, prime forms, or allowing the omission of third element \(c\) of the sequence.
- Identity(Q): QuadBin -> QuadBinElt
- Q ! 1: QuadBin, RngIntElt -> QuadBinElt
Create the principal form in the structure \(Q\) of binary quadratic forms of discriminant \(D\). The principal form is a reduced form equivalent to \(X^2-D/4Y^2\) when \(D \equiv 0 \bmod 4,\) or \(X^2+XY-(D-1)/4Y^2\) when \(D \equiv 1 \bmod 4\).
- Q ! [a, b, c]: QuadBin, RngIntElt, RngIntElt, RngIntElt -> QuadBinElt
- elt< Q | a, b, c>: QuadBin, RngIntElt, RngIntElt, RngIntElt -> QuadBinElt
- elt< Q | a, b>: QuadBin, RngIntElt, RngIntElt -> QuadBinElt
Returns the binary quadratic form \(aX^2+bXY+cY^2\) in the magma of forms \(Q\) of discriminant \(D\). Here \(c\) is determined by the solution of the equality \(D = b^2-4ac\); if no integer \(c\) exists satisfying this, an error will occur.